(b1, b2, ..., bn) be any two vectors in R". The inner product Let a = (a1, a2, ..., an) and b = (dot product) of these two vectors are defined as ā ·b = ajbj + a2b2 + · ·. + anbn, and also the norms of these vectors are defined as ||ā|| = Vā a = Va? + a3 + + an, |||| = V5 .5 = /b? + b3 + + b . Prove the Cauchy-Schwarz inequality (a · b)? < |lā||2|||P', that is the inequality (a¡b1 + azb2 +...+ anbn)² < (a² + a3 + ..+ an)(bỉ + b3 + ·..+ b). + b%). Hint: Consider the function f(x) = (a,x + b1)² + (a2x + b2)² +...+ (anx+bn)² and apply (a).

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